$\sum\limits_{k=1}^{550 }{{(2k + 50)}}=$
Solution: What is the question asking for? The question is asking for the sum of the values of $2k + 50$ from $k = 1$ to $k = 550 $ : $(2 \cdot 1 + 50) + (2 \cdot 2 + 50) +... + (2\cdot {550} +50)$ The series is arithmetic because the formula $2k + 50$ is a linear function of $k$. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The number of terms $(n = {550})$ is the upper limit of the sigma notation. We need to find $a_1$ (the first term) and $a_{550}$ (the last term). Step 1: Find $a_1$ and $a_{550}$ (the first and the last term) $a_1 = 2(1) + 50 = {52}$ $a_{550} = 2(550) + 50 = {1150}$ Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{550}}&= \dfrac {\left({52} + {1150} \right)}{2} \cdot {550} \\\\ S_{{550}} &= 601 \left(550\right) \\\\ S_{{550}} &= 330{,}550\end{aligned}$ The answer $ 330{,}550 $